论文标题
平面随机群集模型：缩放关系
Planar random-cluster model: scaling relations
论文作者
论文摘要
本文使用新型的耦合技术研究了$ \ mathbb z^2 $上的平面随机群集模型的关键和临界机制，并使用新颖的耦合技术在[1,4] $中进行了关键和临界模型。更确切地说，我们得出了关键指数$β$，$γ$，$δ$，$η$，$ν$，$ζ$以及$α$（当$α\ ge0 $）之间的比例关系。作为关键输入，我们使用对混合速率对边缘影响的概念进行了新的解释，显示了在近临界制度中穿越概率的稳定性。作为副产品，我们得出了凯斯滕（Kesten）对伯努利（Bernoulli）渗透的经典缩放关系的概括，涉及``混合率'''关键指数$ $ $替换四臂事件指数$ξ_4$。
This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $β$, $γ$, $δ$, $η$, $ν$, $ζ$ as well as $α$ (when $α\ge0$). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the ``mixing rate'' critical exponent $ι$ replacing the four-arm event exponent $ξ_4$.
